What are the odds of being able to disrupt a spellcaster?

Question

What are the chances that the opponent is able to get their attack in to disrupt a spellcaster as a function of casting time and weapon speed?

Assume the opponent will hit, and use this summary of the initiative rules from the DMG:

1. If the spellcaster lost initiative, their spell will always go after the actions of the winning side and any "blow, missile, or appropriate spell (not saved against or saveable against)" will disrupt the spell.
2. If initiative is tied, then the spell goes first if its casting time is less than the weapon's speed factor, simultaneously if they are the same and after if it is larger.
3. If the spellcaster won the initiative, you subtract the losing initiative roll from the weapon's speed factor and take the absolute value of the result. If that is less than the spell's casting time, the spell can be disrupted, if its the same, the blow and the spell go off simultaneously and if its bigger, the spell goes first.

Answer 1

Overall the results look like this:

which I'd summarize as:

• The probability of disruption is about 1 for weapons that are faster than the casting time
• The probability is about 5/6 for ties (except for 1-2 segment spells),
• The probability is about 1/2 if the spell is faster than the weapon

Raw numbers

C↓ \ W →	1	2	3	4	5	6	7	8	9	10
1	0.56	0.53	0.50	0.47	0.44	0.42	0.42	0.42	0.42	0.42
2	0.83	0.75	0.67	0.58	0.50	0.44	0.42	0.42	0.42	0.42
3	0.92	0.97	0.83	0.69	0.58	0.50	0.44	0.42	0.42	0.42
4	0.97	1.00	1.00	0.83	0.69	0.58	0.50	0.44	0.42	0.42
5	1.00	1.00	1.00	1.00	0.83	0.69	0.58	0.50	0.44	0.42
6	1.00	1.00	1.00	1.00	1.00	0.83	0.69	0.58	0.50	0.44
7	1.00	1.00	1.00	1.00	1.00	1.00	0.83	0.69	0.58	0.50
8	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.83	0.69	0.58
9	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.83	0.69
10	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.83

Methodology

Think of the 6x6 array of possible initiative rolls:

	1	2	3	4	5	6
1	T	D	D	D	D	D
2	1	T	D	D	D	D
3	1	2	T	D	D	D
4	1	2	3	T	D	D
5	1	2	3	4	T	D
6	1	2	3	4	5	T

with the columns corresponding to the initiative roll of the opponent, the rows corresponding to the initiative roll of the caster.

• There are 15 slots marked "D" where the opponent won initiative outright, so will disrupt the spell on a successful attack
• There are 6 slots marked "T" where the opponent can disrupt if the weapon speed factor is less than the casting time
• The rest are marked with numbers, and spell disruption will occur for a successful attack if |W-#| < C where W is the weapon speed, # is the number in the cell (which is the value of the losing initiative roll), and C is the casting time.

So calculating the odds is just counting up the cells and then dividing by 36. For example, for a 2 segment spell, and a weapon speed of 5 (longsword), there are the 15 [opponent won] +2 [opponent rolled 4, but lost initiative] and +1 [opponent rolled 5, but lost initiative] = 18 cells that correspond to initiative rolls that could disrupt the caster; 18/36 = 0.5. Note that due to the weird math involved, the opponent rolling a 1 does not result in spell disruption for the 2 segment casting vs 5 weapon-speed situation.

Note that ties means "no disruption".